From ec0807abd7738788884f8e129bdd8290caa325c0 Mon Sep 17 00:00:00 2001
From: Pauli Virtanen <pauli.t.virtanen@jyu.fi>
Date: Mon, 29 Aug 2022 16:16:02 +0300
Subject: [PATCH] DOC

---
 doc/discretization.md | 54 +++++++++++++------------------------------
 1 file changed, 16 insertions(+), 38 deletions(-)

diff --git a/doc/discretization.md b/doc/discretization.md
index 8f171ed..5b4f913 100644
--- a/doc/discretization.md
+++ b/doc/discretization.md
@@ -8,11 +8,12 @@ in the action formulation.
 
 The action is
 ```math
-  S[Q] = \mathrm{Tr}[D(\hat{\nabla}Q)^2 + F_{ij}Q\hat{\nabla}_iQ\hat{\nabla}_jQ + \Omega Q]
+  S[Q] = \frac{i\pi\nu}{8} \mathrm{Tr}[D(\hat{\nabla}Q)^2 + \alpha F_{ij}Q\hat{\nabla}_iQ\hat{\nabla}_jQ + 4 i \Omega Q]
   \,,
 ```
-where $`\hat{\nabla}_j Q = \partial_j Q - i[\mathcal{A}_j, Q]`$.  We'd
-like the discretized functional to be also gauge invariant.
+where $`\hat{\nabla}_j Q = \partial_j Q - i[\mathcal{A}_j, Q]`$
+and $\Omega = \epsilon\tau_3 + i\Delta$, $\Delta^\dagger = \Delta$.
+We'd like the discretized functional to be also gauge invariant.
 
 We subdivide the space to cells, centered on a rectangular lattice
 $`\vec{r}_j=(h j_x, h j_y, h j_z)`$ with $`j_{x,y,z}\in\mathbb{Z}`$
@@ -49,7 +50,6 @@ the symmetry.
 ## Derivative term
 
 We then discretize
-%
 ```math
 S_2
 =
@@ -78,19 +78,7 @@ U_{ji}
 =
 \sum_{n=0}^\infty i^n |\vec{r}_i-\vec{r}_j|^n \int_{-1/2}^{1/2}ds_1\int_{-1/2}^{s_1}ds_2\ldots\int_{-1/2}^{s_{n-1}}ds_n\, \mathcal{A}(s_1)\cdots\mathcal{A}(s_n)
 ```
-```math
-U_{ji}
-\approx
-  1 + i h \mathcal{A}(s=0)
-  +
-  \frac{(ih)^2}{2}\mathcal{A}(s=0)^2
-  +
-  \mathcal{O}(h^3)
-  \,,
-```
-where $`\mathcal{A}(s)=\mathcal{A}[(\frac{1}{2} - s)\vec{r}_i + (\frac{1}{2} + s)\vec{r}_j]`$. The second line indicates the midpoint rule,
-found using $`\mathcal{A}(s)\simeq\mathcal{A}(\frac{1}{2}) + (s-\frac{1}{2})\mathcal{A}'(\frac{1}{2})+\ldots`$
-and observing that and $`\frac{d^n}{ds^n}\mathcal{A}(s) = \mathcal{O}(h^n)`$.
+where $`\mathcal{A}(s)=\mathcal{A}[(\frac{1}{2} - s)\vec{r}_i + (\frac{1}{2} + s)\vec{r}_j]`$.
 
 As well known in lattice QCD, the field strength can be expressed in
 terms of the link matrices $`U`$ via a plaquette loop.  Consider then
@@ -202,24 +190,16 @@ We can also calculate the matrix current $j\mapsto{}i$ between neighboring cells
   J_{ij}^a
   =
   \frac{\partial}{\partial\xi}
-  S[
-    U_{ij}
-    \mapsto
-    e^{i\xi h T^a} U_{ij}
-  ]\rvert_{\xi=0}
-  =
-  \frac{\partial}{\partial\xi}
   S[
     U_{ij}
     \mapsto
     U_{ij}
     +
-    \frac{ih}{2}T^a U_{ij}\xi
+    T^a U_{ij}\xi
   ]\rvert_{\xi=0}
 ```
-%
 where $`T^a`$ is an appropriate matrix generator, and the transformation
-is inserted only to one of the links.
+is inserted to only one of the links.
 
 The local gauge invariance now implies that there is an exact discrete
 continuity equation
@@ -227,10 +207,10 @@ continuity equation
   \sum_{j\in{}N_i} J_{ij}^a
   =
   \frac{\partial}{\partial\xi}
-  S[Q(i)\mapsto{}e^{i\xi T^a}Q(i)e^{-i\xi T^a}]\rvert_{\xi=0}
+  S[Q(i)\mapsto{}e^{\xi T^a}Q(i)e^{-\xi T^a}]\rvert_{\xi=0}
   =
   \frac{\partial}{\partial\xi}
-  S[\Omega(i)\mapsto{}e^{-i\xi T^a}\Omega(i)e^{i\xi T^a}]\rvert_{\xi=0}
+  S[\Omega(i)\mapsto{}e^{-\xi T^a}\Omega(i)e^{\xi T^a}]\rvert_{\xi=0}
   =
   R_{i}^a
   \,.
@@ -238,19 +218,17 @@ continuity equation
 This is one main advantage of the procedure, in addition to its
 invariance vs. gauge fixing.
 
-However, the definition of the matrix current is \emph{asymmetric}, so
+However, the definition of the matrix current is *asymmetric*, so
 that $`J_{ij}\ne{}-J_{ji}`$ as they differ by a quantity of order
 $`\mathcal{O}(h^2)`$. The asymmetry arises when $`T^a`$ does not commute
 with $`U_{ij}`$, so it is an issue only for the SU(2) component. The
-problem is to some degree just in the interpretation of what $`J_{ij}`$
-means: 
+problem is only in the interpretation.
 
-It is the incoming current measured at site $`i`$, whereas
-$`J_{ji}`$ is measured as site $`j`$.  Since the parallel transport of
-spin between these two locations can imply rotation due to the gauge
-field, there's no reason why we should have $`J_{ij}=-J_{ji}`$, except
-in the continuum limit where the two points become close to each
-other.
+The currents measured at sites $`i`$ and $`j`$ are different: since
+the parallel transport of spin between these two locations can imply
+rotation due to the gauge field, there's no reason why we should have
+$`J_{ij}=-J_{ji}`$, except in the continuum limit where the two points
+become close to each other.
 
 
 ## Implementation
-- 
GitLab